User:Stevelihn/sandbox/Incomplete moment generating function

In probability theory and statistics, the put and call incomplete moment generating function s of a real-valued random variable are defined similarly to the moment generating function (MGF) of a probability distribution, but with different or "incomplete" integral limits. The MGF is defined as an integral from negative infinity to positive infinity. This contrasts with the put incomplete MGF, which is defined as an integral from negative infinity to a variable upper limit. Similarly, the call incomplete MGF is defined as an integral from a variable lower limit to positive infinity. The put and call functions are defined in the same spirit as the lower and upper functions of the incomplete gamma function.

In financial modeling, the option generating functions are derived from the incomplete MGF's, and serve as abstraction of normalized option prices for a given probability distribution in Lihn's option pricing model. The put function is associated with the put option, and the call function is associated with the call option.

Definition
Assume we have a probability distribution of a real-valued random variable $$X$$, whose probability density function (PDF) is $$P(x)$$ and cumulative distribution function (CDF) is $$\Phi(x)$$, $$x\in(-\infty,\infty)$$. Its put and call incomplete moment generating functions are defined as (See Section 3.3 of )


 * $$\begin{align}

M_c(k,t) & = E[e^tX]_{X\geq k} = \int_k^\infty e^{tx} P(x) \, dx; \\[4pt] M_p(k,t) & = E[e^tX]_{X\leq k} = \int_{-\infty}^k e^{tx} P(x) \, dx. \end{align}$$

The parameter $$k$$ represents the logarithm of strike price in option pricing model.

It is obvious that the moment generating function $$M(t)=E[e^{tX}]$$ is simply $$M_c(-\infty,t)$$ or $$M_p(\infty,t)$$. It is also obvious that $$M_c(k,t) + M_p(k,t) = M(t)$$, which is related to the put-call parity.

When $$t=1,$$ we simplify the notations by $$M_c(k) = M_c(k,t=1)$$ and $$M_p(k) = M_p(k,t=1)$$.

Properties
Assume $$P(x) \equiv P(x;\mu,\sigma)$$ is a location-scale distribution family where $$\mu$$ represents the mean, and $$\sigma$$ represents the scale (that is, the standard deviation is proportional to $$\sigma$$). We can parametrize the incomplete MGF as $$M_{c,p}(k,t;\mu,\sigma)$$, and the unit distribution has $$M_{c,p}(k,t;0,1)$$. The following scaling properties hold:


 * $$M_{c,p}(k,t; \mu,\sigma) =

\begin{cases} e^{t\mu} M_{c,p}(k-\mu,t; 0,\sigma), & \text{shifting } \mu; \\[4pt] M_{c,p} \left(\frac{k}{\sigma}, \sigma t; \frac{\mu}{\sigma},1\right), & \text{scaling } \sigma; \\[4pt] e^{t\mu} M_{c,p} \left(\frac{k-\mu}{\sigma}, \sigma t; 0,1\right), & \text{via unit distribution}. \end{cases}$$

The risk-neutral drift $$\mu_D$$ is defined as $$\mu_D=-\log M(t=1;\mu=0)$$.

Option generating function
The put and call option generating functions (OGF) are derived immediately from incomplete MGF's for option pricing modeling. They are defined as


 * $$\begin{align}

L_c(k) & = \int_k^\infty (e^x - e^k) P(x) \, dx; \\[4pt] L_p(k) & = -\int_{-\infty}^k (e^x - e^k) P(x) \, dx. \end{align}$$

In Lihn's option pricing model, $$L_{c,p}(k)$$ represents the normalized option price at log-strike $$k$$  for one unit of underlying security, assuming the option style is European. We can rewrite them in terms of incomplete MGF and CDF, such as


 * $$\begin{align}

L_c(k) & = M_c(k)- e^k (1-\Phi(k)); \\[4pt] L_p(k) & = -M_p(k) + e^k \Phi(x). \end{align}$$

The put-call parity is expressed as
 * $$L_c(k) - L_p(k) = 1 + \Delta \mu - e^k, \text{ where } \Delta \mu=M(t=1)-1.$$

In addition, $$L_{c,p}(k)$$ is translation-invariant, so that


 * $$L_{c,p}(k;\mu,\sigma) = e^\mu L_{c,p}(k-\mu;0,\sigma).$$

Derivation
The price of a call option is typically defined as $$C=\max\{(S-K),0\}$$, and for a put option, $$P=\max\{(K-S),0\}$$, where $$S$$ is the underlying security price and $$K$$ is the strike price. Assume $$S_0$$ is the security price at present and $$S_T$$ is the security price at maturity, whose logarithm is subject to the probability density of $$P(x)$$, we have expected option prices:


 * $$\begin{align}

C & = E[S_T-K]_{S_T>K}; \\[4pt] P & = E[K-S_T]_{S_T<K}. \end{align}$$

Let $$k=\log(K/S_0)$$ and $$x=\log(S_T/S_0)$$, it follows that (See Section 3.3 of )


 * $$\begin{align}

L_c(k) & \equiv \frac{C}{S_0} = \int_k^\infty (e^x - e^k) P(x) \, dx; \\[4pt] L_p(k) & \equiv \frac{P}{S_0} = -\int_{-\infty}^k (e^x - e^k) P(x) \, dx. \end{align}$$

Applications
Two analytical examples are the normal distribution and the Laplace distribution. The case of normal distribution is equivalent to the Black–Scholes option pricing model. The Laplace distribution illustrates how option prices change in a leptokurtic scenario (fat tail). We use $$\,^{(N)}$$superscript for the results from normal distribution below, and $$\,^{(L)}$$ for the results from Laplace distribution.

Normal distribution
Based on the standard parametrization $$\mathcal{N}(\mu,\sigma^2)$$, we have analytic form for the following: $$M^{(N)}(t)=e^{\mu t+\sigma^2t^2/2}$$, thus $$\mu_D^{(N)}=-\sigma^2/2$$. Note that $$\sigma^2$$ represents total variance from now to maturity, thus the Black–Scholes volatility is $$\sigma_{BS} = \frac{\sigma}{\sqrt{T}}$$.

The incomplete MGFs are


 * $$\begin{align}

M^{(N)}_c(k,t) & = \frac{1}{2} e^{\mu+\sigma^2/2} \left[1-\operatorname{erf} \left( \frac{k-\mu}{\sqrt{2}\sigma}-\frac{\sigma}{\sqrt{2}} \right) \right]; \\[4pt] M^{(N)}_p(k,t) & = \frac{1}{2} e^{\mu+\sigma^2/2} \left[1+\operatorname{erf} \left( \frac{k-\mu}{\sqrt{2}\sigma}-\frac{\sigma}{\sqrt{2}} \right) \right]. \end{align}$$

Black–Scholes call option
Black–Scholes model uses $$d_1=\frac{1}{\sigma} \left( -k+ \frac{\sigma^2}{2} \right)$$, $$d_2 = d_1-\sigma = \frac{1}{\sigma} \left( -k- \frac{\sigma^2}{2} \right)$$ and the CDF of standard normal distribution: $$N(x) = \frac{1}{2} \left[ 1 + \operatorname{erf} \left( \frac{x-\mu}{\sigma\sqrt 2} \right) \right] $$ to simplify the notation (risk-free rate is set to zero here). With $$

\operatorname{erf} (-x) = -\operatorname{erf} (x) $$, the call OGF evaluated at $$\mu=\mu_D$$ is



\begin{align} L^{(N)}_c(k;\mu=\mu_D) & = \frac{1}{2} (1-e^k) - \frac{1}{2} \operatorname{erf} \left( \frac{k}{\sigma\sqrt{2}}-\frac{\sigma}{2\sqrt{2}} \right) + \frac{e^k}{2} \operatorname{erf} \left( \frac{k}{\sigma\sqrt{2}}+\frac{\sigma}{2\sqrt{2}} \right) \\[4pt]

& = \frac{1}{2} (1-e^k) - \frac{1}{2} \operatorname{erf} \left( -\frac{d_1}{\sqrt{2}} \right) + \frac{e^k}{2} \operatorname{erf} \left( -\frac{d_2}{\sqrt{2}} \right) \\[4pt]

& = N \left( \frac{d_1}{\sqrt{2}} \right) + N \left( \frac{d_2}{\sqrt{2}} \right) \, e^k.

\end{align} $$

We arrive at the Black–Scholes formula for a call option by noting that $$ e^k = K/S_0 $$ and $$ L^{(N)}_c = C/S_0 $$. To add the risk-free rate back, one can follow the alternative formulation to discount the underlying price $$S_0 \rightarrow e^{rT} S_0$$ and the option price $$C \rightarrow C e^{rT}$$.

Black–Scholes put option
The put OGF can be derived from the put-call parity, by noticing $$\Delta \mu = M(t=1)-1 = 0 \ \text{when } \mu=\mu_D,$$



\begin{align} L^{(N)}_p(k;\mu=\mu_D) & = L^{(N)}_c(k;\mu=\mu_D) - (1 - e^k) \\[4pt] & = -\frac{1}{2} ( 1-e^k) - \frac{1}{2} \operatorname{erf} \left( -\frac{d_1}{\sqrt{2}} \right) + \frac{e^k}{2} \operatorname{erf} \left( -\frac{d_2}{\sqrt{2}} \right) \\[4pt]

& = - N \left( -\frac{d_1}{\sqrt{2}} \right) + N \left( -\frac{d_2}{\sqrt{2}} \right) \, e^k.

\end{align} $$ We arrive at the Black–Scholes formula for a put option.

Laplace distribution
We use parametrization of PDF, $$P^{(L)}(x) = \frac{1}{2\sigma} e^{-|x-\mu|/\sigma}$$, we have analytic form for $$M^{(L)}(t)=e^{\mu t} (1-\sigma^2t^2)^{-1}$$, thus $$\mu_D^{(L)}=\log(1-\sigma^2)$$, where $$|\sigma t|<1$$. Define $$B^{\pm} = 1 \pm \sigma$$ and $$\hat{k} = \frac{k-\mu}{\sigma}$$, we have


 * $$\begin{align}

M^{(L)}_c(k,t) & = \frac{1}{2 B^-} e^{\mu -B^- \left|\hat{k}\right|} , & \text{when } \hat{k} \geq 0; \\[6pt] M^{(L)}_p(k,t) & = \frac{1}{2 B^+} e^{\mu -B^+ \left|\hat{k}\right|} , & \text{when } \hat{k} < 0. \end{align}$$

With $$\hat{k}^{(L)}_D = \frac{k-\mu^{(L)}_D}{\sigma}$$, the Laplace OGFs are (See Section 3.8.2 of )


 * $$\begin{align}

L^{(L)}_c(k;\mu=\mu^{(L)}_D) & = \frac{\sigma}{2 B^-} e^{\mu -B^- \left|\hat{k}^{(L)}_D\right|} , & \text{when } \hat{k}^{(L)}_D \geq 0; \\[4pt] L^{(L)}_p(k;\mu=\mu^{(L)}_D) & = \frac{\sigma}{2 B^+} e^{\mu -B^+ \left|\hat{k}^{(L)}_D\right|} , & \text{when } \hat{k}^{(L)}_D < 0. \end{align}$$

The put-call parity, $$L^{(L)}_c(k) - L^{(L)}_p(k) = 1-e^k$$, is used to derive the other half of the solutions.

Implied volatility of a non-normal probability distribution
Take the call option from a Laplace distribution as an example, the Black–Scholes implied volatility is calculated as



\sigma_{BS}(k) = \frac{1}{\sqrt{T}} \left[ L^{(N)}_c \right]^{-1} \left( L^{(L)}_c(k;\mu=\mu^{(L)}_D) \right), $$ where $$ \left[ L^{(N)}_c \right]^{-1}(\cdot) $$ is the inverse function of $$ L^{(N)}_c(k;\mu=\mu_D) $$ that finds $$ \sigma $$ given a normalized option price from OGF. The $$k$$ dependency of $$\sigma_{BS}(k)$$ is an illustration of the volatility smile.

The following R script based on the ecd package calculates such volatility smile: library(ecd) ttm = 1/12 # one month sigma = 0.15 / sqrt(2) * sqrt(ttm) # approx 15% volatility ld = ecld(lambda=2, sigma=sigma) # lambda=2, Laplace distribution # k = seq(-0.1, 0.1, length.out=60) # range of log-strike px = ecld.ogf(ld, k) bs_vol = ecld.op_V(px, k, ttm=ttm) # par(mfcol=c(2,1)) plot(k, px, type="l", col="blue") plot(k, bs_vol, type="l", col="blue")